Spin as a consequence of a multi-valued phase

8 Spin as a consequence of a multi-valued phase

As shown by [5], [19], and others, spin in non-relativistic QT may be introduced in exactly the same manner as the electrodynamic potentials. In this section we shall apply a slightly modified version of their method and try to derive spin in an alternative way - which avoids the shortcoming mentioned in the last section.

[5] and [19] introduce the potentials by applying the well-known minimal-coupling rule to the free Hamiltonian. In the present treatment this is achieved by making the quantity $S$ multi-valued. The latter approach seems intuitively preferable considering the physical meaning of the corresponding classical quantity. Let us first review the essential steps [see [32] for more details] in the process of creating potentials in the scalar Schrödinger equation:

Chose a free Schrödinger equation with single-valued state function.
’Turn on’ the interaction by making the state function multi-valued (multiply it with a multi-valued phase factor)
Shift the multi-valued phase factor to the left of all differential operators, creating new terms (potentials) in the differential equation.
Skip the multi-valued phase. The final state function is again single-valued.

Let us adapt this method for the derivation of spin (considering spin one-half only). The first and most important step is the identification of the free Pauli equation. An obvious choice is
$[ ] ℏ--∂-- --1--- ℏ---∂--- 2 ¯ + ( ) + V ψ = 0, i ∂ t 2m i ∂ ⃗x$ (96)

where $¯ ψ$ is a single-valued two-component state function; (96) is essentially a duplicate of Schrödinger’s equation. We may of course add arbitrary vanishing terms to the expression in brackets. This seems trivial, but some of these terms may vanish only if applied to a single-valued $ψ¯$ and may lead to non-vanishing contributions if applied later (in the second of the above steps) to a multi-valued state function $ψ¯multi$ .

In order to investigate this possibility, let us rewrite Eq. (96) in the form
$[ ] --1---⃗ ⃗ ¯ ˆp0 + ˆp ˆp + V ψ = 0, 2m$ (97)

where $ˆp0$ is an abbreviation for the first term of (96) and the spatial derivatives are given by
$ℏ ∂ ⃗ˆp = ˆpk ⃗ek , ˆpk = ---------. i ∂ x k$ (98)

All terms in the bracket in (97) are to be multiplied with a $2x2$ unit-matrix $E$ which has not be written down. Replace now the derivatives in (97) according to
$pˆ ⇒ ˆp M , ⃗ˆp ⇒ ⃗ˆp M , 0 0 0 k k$ (99)

where $M , M 0 k$ are hermitian $2x2$ matrizes with constant coefficients, which should be constructed in such a way that the new equation agrees with (97) for single-valued $ψ¯$ , i.e. assuming the validity of the condition
$(pˆ ˆp - pˆ ˆp ) ψ¯ = 0. i k k i$ (100)

This leads to the condition
$- 1 M 0 MiMk = E δik + Tik ,$ (101)

where $Tik$ is a $2x2$ matrix with two cartesian indices $i, k$ , which obeys $Tik = - Tki$ . A solution of (101) is given by $M = σ , M = σ 0 0 i i$ , where $σ , σ 0 i$ are the four Pauli matrices. In terms of this solution, Eq. (101) takes the form
$σi σk = σ0 δik + i εikl σl.$ (102)

Thus, an alternative free Pauli-equation, besides (96) is given by
$[ ] ( )2 ℏ ∂ 1 ℏ ∂ ∂ ------ + ------ --- σ -----σ -------+ V ψ¯ = 0. i k i ∂ t 2m i ∂ xi ∂ xk$ (103)

The quantity in the bracket is the generalized Hamiltonian constructed by [5] and [19]. In the present approach gauge fields are introduced by means of a multi-valued phase. This leads to the same formal consequences as the minimal coupling rule but allows us to conclude that the second free Pauli equation (103) is more appropriate than the first, Eq. (96), because it is more general with regard to the consequences of multi-valuedness. This greater generality is due to the presence of the second term on the r.h.s. of (102).

The second step is to turn on the multi-valuedness in Eq. (103), $¯ ¯ multi ψ ⇒ ψ$ , by multiplying $¯ ψ$ with a multi-valued two-by-two matrix. This matrix must be chosen in such a way that the remaining steps listed above lead to Pauli’s equation (93) in presence of an gauge field. Since in our case the final result (93) is known, this matrix may be found by performing the inverse process, i.e. performing a singular gauge transformation $ˆ ¯ multi ψ = Γ ψ$ of Pauli’s equation (93) from $ˆ ψ$ to $¯ multi ψ$ , which removes all electrodynamic terms from (93) and creates Eq. (103). The final result for the matrix $Γ$ is given by
$∫ x,t [ ] -e-- ′ ′ ′ ′ ′ ′ Γ = E exp {i dx kAk (x , t ) - cdt ϕ (x , t ) }, ℏc$ (104)

and agrees, apart from the unit matrix $E$ , with the multi-valued factor introduced previously [see (17) and (52)] leading to the electrodynamic potentials. The inverse transition from (103) to (93), i.e. the creation of the potentials and the Zeeman term, can be performed by using the inverse of (104).

The Hamiltonian (103) derived by [5] and [19] shows that spin can be described by means of the same abelian gauge theory that leads to the standard quantum mechanical gauge coupling terms; no new adjustable fields or parameters appear. The only requirement is that the appropriate free Pauli equation (103) is chosen as starting point. The theory of [12], on the other hand, started from the alternative (from the present point of view inappropriate) free Pauli equation (96) and leads to the conclusion that spin must be described by a non-abelian gauge theory.

As far as our derivation of non-relativistic QT is concerned we have now two alternative, and in a sense complementary, possibilities to introduce spin. The essential step in the second (Arunsalam-Gould) method is the transition from (96) to the equivalent free Pauli equation (103). This step is a remarkable short-cut for the complicated calculations, performed in the last section, leading to the various terms required by the principle of minimal Fisher information. The Arunsalam-Gould method is unable to provide the shape (95) of the corresponding macroscopic forces but is very powerful insofar as no adjustable quantities are required. It will be used in the next section to perform the transition to an arbitrary number of particles.

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